Artificial neural networks in chemometrics: History, examples and perspectives

Artificial neural networks (ANNs) are non-linear computational tools suitable to a great host of practical application due to their flexibility and adaptability. However, their application to the resolution of chemometric problems is relatively recent (early 1990s).In this communication, different artificial neural networks architectures are presented and their application to different kinds of chemometric problems (mainly classification and regression) is discussed by means of examples taken from the authors' experience, stressing the pros and cons of ANNs with respect to traditional chemometric techniques.     

Least-squares Galerkin procedures for parabolic integro-differential equations

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H(div;Ω)×H¹(Ω) and (L²(Ω))²×L²(Ω), respectively.     

直线上时间随机环境下随机游动的渐近性质

在状态空间是可数情形下,本文给出了时间随机环境下随机游动的一个一般模型.随后,在环境是独立同分布情形下得到了直线上时间随机环境下紧邻随机游动的一个常返与暂留准则和强大数定律;最后讨论了其中心极限定理,它类似与简单随机游动的相应结果.     

The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations

In this paper we establish the existence of a positive solution to a singular coupled system of nonlinear fractional differential equations. Our analysis rely on a nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

On polyhedral retracts and compactifications of locally symmetric spaces

The results in this paper are based on a previously constructed exhaustion of a locally symmetric space V=ΓX by Riemannian polyhedra, i.e., compact submanifolds with corners: V=_s0V(s). We show that the interior of every polyhedron V(s) is homeomorphic to V. The universal covering space X(s) of V(s) is quasi-isometric to the discrete group Γ. It can be written as the complement of a Γ-invariant union of horoballs in X (which in general have intersections giving rise to the corners). This yields exponential isoperimetric inequalities for Γπ₁(V(s)). We also discuss the relation of this compactification of V with the Borel–Serre compactification.     

Graphs with no M-alternating path between two vertices

Let G be a graph with a perfect matching M. In this paper, we prove two theorems to characterize the graph G in which there is no M-alternating path between two vertices x and y in G.     

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

For an open subset U of a locally convex space E, let (H(U),τ₀) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U),τ₀) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.     

Optimal checkpointing intervals of three error detection schemes by a double modular redundancy

This paper considers three checkpointing schemes which combine a double modular redundancy and three types of checkpoints: compare-and-store-checkpoint (CSCP), store-checkpoint (SCP), and compare-checkpoint (CCP). An execution time of a task is divided equally into n intervals, and at the end of each interval, a CSCP is always placed. Further, each CSCP interval is also divided equally into m intervals, and at the end of each interval, either CCP or SCP is placed except the last one. Introducing the overheads of comparison, storage, and retry, the mean execution times to complete a task for three schemes are obtained, using the theory of probability. Optimal checkpointing intervals, which minimize the mean times, are analytically derived, and are numerically computed. Three schemes are compared as numerical examples and the best checkpointing scheme is chosen.     

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.     

Miscellaneous classification results for 2-designs

An orderly algorithm combined with clique searching is used to show that there are—up to isomorphism, in all cases—325,062 resolvable 2-(16,4,2) designs with 339,592 resolutions, 19,072,802 2-(13,6,5) designs, and 15,111,019 2-(14,7,6) designs. Properties of the classified designs are further discussed.